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Mirrors > Home > MPE Home > Th. List > brtrclfv | Structured version Visualization version GIF version |
Description: Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.) |
Ref | Expression |
---|---|
brtrclfv | ⊢ (𝑅 ∈ 𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trclfv 14348 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}) | |
2 | 1 | breqd 5068 | . 2 ⊢ (𝑅 ∈ 𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ 𝐴∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝐵)) |
3 | brintclab 14349 | . . 3 ⊢ (𝐴∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 〈𝐴, 𝐵〉 ∈ 𝑟)) | |
4 | df-br 5058 | . . . . 5 ⊢ (𝐴𝑟𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑟) | |
5 | 4 | imbi2i 337 | . . . 4 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵) ↔ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 〈𝐴, 𝐵〉 ∈ 𝑟)) |
6 | 5 | albii 1811 | . . 3 ⊢ (∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵) ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 〈𝐴, 𝐵〉 ∈ 𝑟)) |
7 | 3, 6 | bitr4i 279 | . 2 ⊢ (𝐴∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)) |
8 | 2, 7 | syl6bb 288 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∈ wcel 2105 {cab 2796 ⊆ wss 3933 〈cop 4563 ∩ cint 4867 class class class wbr 5057 ∘ ccom 5552 ‘cfv 6348 t+ctcl 14333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-trcl 14335 |
This theorem is referenced by: brcnvtrclfv 14351 brtrclfvcnv 14352 trclfvcotr 14357 |
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